Scilab Function
Last update : 12/1/2004

dasrt - DAE solver with zero crossing

### Calling Sequence

  `[r,nn,[,hd]]=dasrt(x0,t0,t [,atol,[rtol]],res [,jac],ng, surf [,info] [,hd])`

### Parameters

- `x0`: is either `y0` (`ydot0` is estimated by `dassl` with zero as first estimate) or the matrix `[y0 ydot0]`. `g(t,y0,ydot0)` must be equal to zero. If you only know an estimate of `ydot0` set `info(7)=1`
  - `y0`: real column vector of initial conditions.
  - `ydot0`: real column vector of the time derivative of `y` at `t0` (may be an estimate).
- `t0`: real number is the initial instant.
- `t`: real scalar or vector. Gives instants for which you want the solution. Note that you can get solution at each dassl's step point by setting `info(2)=1`.
- `nn`: a vector with two entries `[times num]``times` is the value of the time at which the surface is crossed, `num` is the number of the crossed surface
- `atol,rtol`: real scalars or column vectors of same size as `y`. `atol,rtol` give respectively absolute and relative error tolerances of solution. If vectors the tolerances are specified for each component of `y`.
- `res`: external (function or list or string). Computes the value of `g(t,y,ydot)`.
  - `function`

    : Its calling sequence must be `[r,ires]=res(t,y,ydot)` and `res` must return the residue `r=g(t,y,ydot)` and error flag `ires`. `ires = 0` if `res` succeeds to compute `r`, `=-1` if residue is locally not defined for `(t,y,ydot)`, `=-2` if parameters are out of admissible range.

  - `list`: it must be as follows:

        list(res,x1,x2,...)

    where the calling sequence of the function `res` is now

        r=res(t,y,ydot,x1,x2,...)

    `res` still returns `r=g(t,y,ydot)` as a function of `(t,y,ydot,x1,x2,...)`.

  - `string`: it must refer to the name of a fortran subroutine (see source code of `fresd.f`).
- `jac`: external (function or list or string). Computes the value of `dg/dy+cj*dg/dydot` for a given value of parameter `cj`
  - `function`: Its calling sequence must be `r=jac(t,y,ydot,cj)` and the `jac` function must return `r=dg(t,y,ydot)/dy+cj*dg(t,y,ydot)/dydot` where `cj` is a real scalar
  - `list`: it must be as follows

        list(jac,x1,x2,...)

    where the calling sequence of the function `jac` is now

        r=jac(t,y,ydot,x1,x2,...)

    `jac` still returns `dg/dy+cj*dg/dydot` as a function of `(t,y,ydot,cj,x1,x2,...)`.

  - `character string`: it must refer to the name of a fortran subroutine (see source code of `jacdd.f`).
- `surf`: external (function or list or string). Computes the value of the column vector `surf(t,y)` with ng components. Each component defines a surface.
  - `function`: Its calling sequence must be `surf(t,y)`
  - `list`: it must be as follows

        list(surf,x1,x2,...)

    where the calling sequence of the function `surf` is now

        r=surf(t,y,x1,x2,...)

  - `character string`: it must refer to the name of a fortran subroutine (see source code of `fsurfd.f`) in directory `SCDIR/default`
- `info`: list which contains `7` elements, default value is list([],0,[],[],[],0,0)
  - `info(1)` : real scalar which gives the maximum time for which `g` is allowed to be evaluated or an empty matrix `[]` if no limits imposed for time.
  - `info(2)` : flag which indicates if `dassl` returns its intermediate computed values (`flag=1`) or only the user specified time point values (`flag=0`).
  - `info(3)` : `2` components vector which give the definition `[ml,mu]` of band matrix computed by `jac`; `r(i - j + ml + mu + 1,j) = "dg(i)/dy(j)+cj*dg(i)/dydot(j)"`. If `jac` returns a full matrix set `info(3)=[]`.
  - `info(4)` : real scalar which gives the maximum step size. Set `info(4)=[]` if no limitation.
  - `info(5)` : real scalar which gives the initial step size. Set `info(4)=[]` if not specified.
  - `info(6)` : set `info(6)=1` if the solution is known to be non negative, else set `info(6)=0`.
  - `info(7)` : set `info(7)=1` if `ydot0` is just an estimation, `info(7)=0` if `g(t0,y0,ydot0)=0`.
- `hd`: real vector which allows to store the `dassl` context and to resume integration
- `r`: real matrix . Each column is the vector [t;x(t);xdot(t)] where t is time index for which the solution had been computed

### Description

Solution of the implicit differential equation

        g(t,y,ydot)=0
        y(t0)=y0  and   ydot(t0)=ydot0

Returns the surface crossing instants and the number of the surface reached in `nn`.

Detailed examples can be found in SCIDIR/tests/dassldasrt.tst

### Examples

    //dy/dt = ((2*log(y)+8)/t -5)*y,  y(1) = 1,  1<=t<=6
    //g1 = ((2*log(y)+8)/t - 5)*y
    //g2 = log(y) - 2.2491
    y0=1;t=2:6;t0=1;y0d=3;
    atol=1.d-6;rtol=0;ng=2;

    deff('[delta,ires]=res1(t,y,ydot)','ires=0;delta=ydot-((2*log(y)+8)/t-5)*y')
    deff('[rts]=gr1(t,y)','rts=[((2*log(y)+8)/t-5)*y;log(y)-2.2491]')

    [yy,nn]=dasrt([y0,y0d],t0,t,atol,rtol,res1,ng,gr1);
    //(Should return nn=[2.4698972 2])

### See Also

`ode`, `dassl`, `impl`, `fort`, `link`, `external`,